Under the assumption of \(\mathcal {F}_n(\mathsf {P}_0)<\infty \)  we have by Theorem REF  a chain rule for the entropy, the inequality \(\mathcal {I}_n\ge 0\) , and the statement that \(\mathcal {I}_n(\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)=0\)  implies that \(\mathsf {P}\)  is a weak solution. Moreover, due to Theorem REF  there exists a weak solution with \(\mathcal {I}_n\le 0\) .
It remains to show that gradient-flow solutions are unique, which is a classical argument using the strict convexity of \(\mathcal {F}_n\) , e.g. see Theorem 5.9 of [1]}. Suppose that there exist two curves \(\mathsf {P}^1,\mathsf {P}^2\)  such that \(\mathsf {P}^1_0=\mathsf {P}^2_0=\bar{\mathsf {P}}\) , \(\mathcal {I}_n(\mathsf {P}^1,\vartheta _{\mathsf {P}^1}^+,\vartheta _{\mathsf {P}^1}^-)\)  and \(\mathcal {I}_n( \mathsf {P}^2,\vartheta _{\mathsf {P}^2}^+,\vartheta _{\mathsf {P}^2}^-)=0\) . Applying the chain rule it is straightforward to verify that for a gradient-flow solution \(\mathcal {I}_{n}^t=0\)  for every \(t \in [0,T]\) , where
\( \mathcal {I}_n^t(\mathsf {P},\mathsf {J}^+,\mathsf {J}^-):= \int _0^t \mathcal {R}_n(\mathsf {P}_r,\mathsf {J}^+_r,\mathsf {J}_r^-)\, \mathrm {d}r + \mathcal {F}_n(\mathsf {P}_t)-\mathcal {F}_n(\bar{\mathsf {P}})+\int _0^t \mathcal {D}_n(\mathsf {P}_r) \, \mathrm {d}r,\) 
and that \(\mathcal {I}_n^t\ge 0\)  for arbitrary curves with initial condition \(\bar{\mathsf {P}}\) .
Now, define \(\tilde{\mathsf {P}}_t=\genfrac{}{}{}2{1}{2}\mathsf {P}^1+\genfrac{}{}{}2{1}{2}\mathsf {P}^2\)  and note that \((\tilde{\mathsf {P}},\vartheta _{\tilde{\mathsf {P}}}^+,\vartheta _{\tilde{\mathsf {P}}}^-)\in \mathsf {CE}_n\)  as well, and
\( \vartheta _{\tilde{\mathsf {P}}}^{\pm }=\genfrac{}{}{}2{1}{2}\vartheta _{\mathsf {P}^1}^{\pm }+\genfrac{}{}{}2{1}{2}\vartheta _{\mathsf {P}^2}^{\pm }. \) 
Fix any \(t \in [0,T]\)  and suppose that \(\mathsf {P}_t^1\ne \mathsf {P}_t^{2}\) . Then by convexity of \(\mathcal {R}_n\)  and \(\mathcal {D}_n\) , and strict convexity of \(\mathcal {F}_n\) , we have
\(\mathcal {I}_n^t(\tilde{\mathsf {P}},\vartheta _{\tilde{\mathsf {P}}}^{+},\vartheta _{\tilde{\mathsf {P}}}^{-})&= \int _0^t \mathcal {R}_n(\tilde{\mathsf {P}}_r,\vartheta _{\tilde{\mathsf {P}}_r}^+,\vartheta _{\tilde{\mathsf {P}}_r}^-)\, \mathrm {d}r + \mathcal {F}_n(\tilde{\mathsf {P}}_t)-\mathcal {F}_n(\bar{\mathsf {P}})+\int _0^t \mathcal {D}_n(\mathsf {P}_r) \, \mathrm {d}r\\&< \genfrac{}{}{}2{1}{2}\mathcal {I}_n^t(\mathsf {P}^1,\vartheta _{\mathsf {P}^1}^+,\vartheta _{\mathsf {P}^1}^-)+\genfrac{}{}{}2{1}{2}\mathcal {I}_n^t(\mathsf {P}^2,\vartheta _{\mathsf {P}^2}^+,\vartheta _{\mathsf {P}^2}^-)=0,\) 

which leads to a contradiction, and hence \(\mathsf {P}_t^1= \mathsf {P}_t^{2}\)  for all \(t \in [0,T]\) .
